Optimal. Leaf size=79 \[ \frac {\sqrt {a+(b \sin (e+f x)+c \cos (e+f x))^2} E\left (e+f x+\tan ^{-1}(b,c)|-\frac {b^2+c^2}{a}\right )}{f \sqrt {\frac {(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1}} \]
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Rubi [F] time = 0.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {(c+b x)^2}{1+x^2}}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {i \sqrt {a+\frac {(c+b x)^2}{1+x^2}}}{2 (i-x)}+\frac {i \sqrt {a+\frac {(c+b x)^2}{1+x^2}}}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {(c+b x)^2}{1+x^2}}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {i \operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {(c+b x)^2}{1+x^2}}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ \end {align*}
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Mathematica [B] time = 1.67, size = 325, normalized size = 4.11 \[ -\frac {\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right ) \sqrt {2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\left (b^2-c^2\right ) \cos (2 (e+f x))-2 b c \sin (2 (e+f x))+\sqrt {\left (b^2+c^2\right )^2}}{\sqrt {\left (b^2+c^2\right )^2}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {\left (b^2+c^2\right )^2}}{b^2+c^2+2 a+\sqrt {\left (b^2+c^2\right )^2}}\right )}{\sqrt {2} f \sqrt {\left (b^2+c^2\right )^2} \sqrt {\frac {\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right )^2}{\left (b^2+c^2\right )^2}} \sqrt {\frac {2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2}{2 a+\sqrt {\left (b^2+c^2\right )^2}+b^2+c^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {2 \, b c \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (b^{2} - c^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.68, size = 4067972, normalized size = 51493.32 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+{\left (c\,\cos \left (e+f\,x\right )+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \left (b \sin {\left (e + f x \right )} + c \cos {\left (e + f x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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